Formal proof that every purely reactive agent has behaviorally equivalent standard agent
I recommend you to look into the literature about simulation and bisimulation in Automata Theory and its applications to model checking (where you want to make quite regularly proofs of "behavioural equivalence"). One article that discusses this in the context of a technique for model checking known as "Abstraction and Abstraction Refinement" is

Abstraction and Abstraction Refinement
Dennis Dams and Orna Grumberg
In Springer's Handbook of Model Checking, 2018, Chapter 13, pages 385-420


A good (I use it regularly) book that covers behavioural equivalence for a wide variety of automata is 

Verification and Control of Hybrid Systems: A Symbolic Approach
Paulo Tabuada
Springer, 2009